Parametric Equations; Polar Equations

9.2 Assess Your Understanding
Printed Page 653

Concepts and Vocabulary

Multiple Choice Let $C$ denote a curve represented by the parametric equations $x=x\,(t),\quad \ y=y(t),\quad \ a\leq t\leq b$ , where each function $x(t) \,$ and $y(t)$ is continuous on the closed interval $[ a,b]$ and differentiable on the open interval $( a,b)$ . If both $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ are continuous and never simultaneously $0$ on $( a,b)$ , then $C$ is called a [(a) smooth, (b) differentiable, (c) parametric] curve.

(a)

True or False If $C$ is a smooth curve, represented by the parametric equations $x=x(t)$ and $y=y(t),$ $a\leq t\leq b$ , then the slope of the tangent line to $C$ at the point $( x, y)$ is given by the formula $\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt} }{\dfrac{dx}{dt}},$ provided $\dfrac{dx}{dt}\neq 0$ .

True

If in the formula for the slope of a tangent line, $\dfrac{dy}{dx}=0$ $\bigg(but \dfrac{dx}{dt}\neq 0\bigg),$ then the curve has a(n) ______________ tangent line at the point $( x(t) ,y(t) ) .$ If $\dfrac{dx}{dt}=0$ $\bigg(but \dfrac{dy}{dt}\neq 0 \bigg)$ , then the curve has a(n) ______________ tangent line at the point $( x(t) ,y(t) ) .$

Horizontal; vertical

True or False For a smooth curve $C$ represented by the parametric equations $x=x(t),\quad \ y=y(t),$ $a\leq t\leq b,$ the length $s$ of $C$ from $t=a$ to $t=b$ is given by the formula $s=\int_{a}^{b}\sqrt{ \dfrac{d^{2}x}{dt^{2}}+\dfrac{d^{2}y}{dt^{2}}}~dt$ .

False

Skill Building

In Problems 5–12, find $\dfrac{dy}{dx}.$ Assume $\dfrac{dx}{dt}\neq 0$ .

$x(t) =e^{t}\cos t,\;y(t) =e^{t}\sin t$

$\dfrac{dy}{dx}=\dfrac{\sin t+\cos t}{\cos t-\sin t}$

$x(t) =1+e^{-t},\;y(t) =e^{3t}$

$x(t) =t+\dfrac{1}{t},\;y(t) =4+t$

$\dfrac{dy}{dx}=\dfrac{1}{1-\dfrac{1}{t^2}}$

$x(t) =t+\dfrac{1}{t},\;y(t) =t-\dfrac{1}{t}$

$x(t) = \cos t+t\sin t,\;y(t) = \sin t-t\cos t$

$\dfrac{dy}{dx}=\tan t$

$x(t) =\cos ^{3}t,\;y(t) =\sin ^{3}t$

$x(t) =\cot ^{2}t,\;y(t) =\cot t$

$\dfrac{dy}{dx}=\dfrac{1}{2\cot t}$

$x(t) =\sin t,\;y(t) =\sec ^{2}t$

In Problems 13–26, for each pair of parametric equations:

(a) Find an equation of the tangent line to the curve at the given number.

(b) Graph the curve and the tangent line.

654

$x(t) =2t^{2},\quad y(t) =t\quad at\quad t=2$

$y=\dfrac{1}{8}x+1$
(b)

$x(t) =t,\quad y(t) =3t^{2}\quad at\quad t=-2$

$x(t) =3t,\quad y(t) =2t^{2}-1\quad at\quad t=1$

$y=\dfrac{4}{3}x-3$
(b)

$x(t) =2t,\quad y(t) =t^{2}-2\quad at\;t=2$

$x(t) =\sqrt{t},\quad y(t) =\dfrac{1}{t}\quad at\;t=4$

$y=-\dfrac{1}{4}x+\dfrac{3}{4}$
(b)

$x(t) =\dfrac{2}{t^{2}},\quad y(t) =\dfrac{1}{t}\quad at\;t=1$

$x(t) =\dfrac{t}{t+2},\quad y(t) =\dfrac{4}{t+2}\quad at\;t=0$

$y=-2x+2$
(b)

$x(t) =\dfrac{t^{2}}{1+t},\quad y(t) =\dfrac{1}{1+t}\quad at\;t=0$

$x(t) =e^{t},\quad y(t) =e^{-t}\quad at\;t=0$

$y=-x+2$
(b)

$x(t) =e^{2t},\quad y(t) =e^{t}\quad at\;t=0$

$x(t) =\sin t,\quad y(t) =\cos t\quad at\;t=\dfrac{\pi }{4}$

$y=-x+\sqrt{2}$
(b)

$x(t) =\sin ^{2}t,\quad y(t) =\cos t\quad at\;t=\dfrac{\pi }{4}$

$x(t) =4\sin t,\quad y(t) =3\cos t\quad at\;t=\dfrac{\pi }{3}$

$y=-\dfrac{3\sqrt{3}}{4}x+6$
(b)

$x(t) =2\sin t-1,\quad y(t) =\cos t+2\quad at\;t=\dfrac{\pi }{6}$

In Problems 27–30, for each smooth curve, find any points where the tangent line is either horizontal or vertical.

$x(t)=t^2,\quad y(t)=t^3-4t$

Horizontal at $\left(\dfrac{4}{3},-\dfrac{16\sqrt{3}}{9}\right)$ , $\left(\dfrac{4}{3},\dfrac{16\sqrt{3}}{9}\right)$ ; vertical at $(0,0)$

$x(t) = t^3 -9t ,\quad y(t)=t^2$

$x(t) =1-\cos t,\quad y(t) =1-\sin t,\quad 0\leq t \leq 2\pi$

Horizontal at $(1,0)$ , $(1,2)$ ; vertical at $(0,1)$ , $(2,1)$

$x(t) =-3\cos t+\cos (3t),\quad y(t) =\sin t,\quad 0\leq t\leq 2\pi$

In Problems 31–38, find the length of each curve over the given interval.

$x(t)=t^{3},\quad y(t)=t^{2};\quad 0\leq t\leq 2$

$\dfrac{8}{27}\big(10\sqrt{10}\,-1\big)$

$x(t)=3t^{2}+1,\quad y(t)=t^{3}-1;\quad 0\leq t\leq 2$

$x(t)=t-1,\quad \ y(t)=\dfrac{1}{2}t^{2};\quad 0\leq t\leq 2$

$s = \sqrt{5} + \dfrac{1}{2} \ln (2 + \sqrt{5})$

$x(t)=t^{2},\quad y(t)=2t;\;1\leq t\leq 3$

$x(t)=4\sin t,\quad y(t)=4\cos t;\; -\dfrac{\pi }{2}\leq t\leq \dfrac{\pi }{2}$

$4\pi$

$x(t)=6\sin t,\quad y(t)=6\cos t;\; -\dfrac{\pi }{2}\leq t\leq \dfrac{\pi }{2}$

$x(t)=2\sin t-1,\quad y(t)=2\cos t+1;\quad 0\leq t\leq 2\pi$

$4\pi$

$x(t)=e^{t}\sin t,\quad y(t)=e^{t}\cos t;\quad 0\leq t\leq \pi$

In Problems 39–44:

(a) Use the arc length formula for parametric equations to set up the integral for finding the length of each curve over the given interval.

(b) Find the length $s$ of each curve over the given interval.

(c) Graph each curve over the given interval.

$x(t) =2\cos ( 2t) ,\quad y(t) =t^{2};\;0\leq t\leq 2\pi$

$s= \int^{2\pi}_{0}\sqrt{[-4\ \sin(2t)]^2+(2t)^2}dt$
$s\approx44.527$
(c)

$x(t) =t^{2},\quad y(t) =\sin t;\quad 0\leq t\leq 2\pi$

$x(t) =t^{2},\quad y(t) =\sqrt{t+2};\;-2\leq t\leq 2$

$s= \int^{2}_{-2}\sqrt{(2t)^2+\left(\dfrac{1}{2\sqrt{t+2}}\right)^2}dt$
$s\approx8.429$
(c)

$x(t) =t-\cos t,\quad y(t) =1-\sin t;\quad 0\leq t\leq \pi$

$x(t) =3\cos t+\cos \left( 3t\right) ,\quad y(t) =3\sin t-\sin ( 3t);\;0\leq t\leq 2\pi$ (a hypocycloid)

$s=4\cdot\int^{\pi/2}_{0}\sqrt{(-3\ \sin t-3\ \sin(3t))^2+(3\ \cos t-3\ \cos(3t))^2}\,dt$
$s=24$
(c)

$x(t) =5\cos t-\cos ( 5t) ,\quad y(t) =5\sin t-\sin ( 5t) ;\; 0\leq t\leq 2\pi$ (an epicycloid)

Applications and Extensions

Tangent Lines
1. Find all the points on the plane curve $C$ represented by $x(t) =t^{2}+2,\quad y(t) =t^{3}-4t$ , where the tangent line is horizontal or where it is vertical.
2. Show that $C$ has two tangent lines at the point $( 6,0) .$
3. (c) Find equations of these tangent lines.
4. Graph $C$ .

Horizontal at $\left(\dfrac{10}{3},-\dfrac{16\sqrt{3}}{9}\right)$ , $\left(\dfrac{10}{3},\dfrac{16\sqrt{3}}{9}\right)$ ; vertical at $(2,0)$
$t=2$ , $t=-2$ ; see Student Solutions Manual
$y=2x-12$ , $y=-2x+12$
(d)

Tangent Lines
1. Find all the points on the plane curve $C$ represented by $x(t) =2t^{2},\quad \ y(t) =8t-t^{3}$ , where the tangent line is horizontal or where it is vertical.
2. Show that $C$ has two tangent lines at the point $( 16,0)$ .
3. (c) Find equations of these tangent lines.
4. Graph $C$ .

Arc Length
1. Find the arc length of one arch of the four-cusped hypocycloid $x(t) =b\sin ^{3}t,\quad y(t) =b\cos ^{3}t,\quad 0\leq t\leq \dfrac{\pi }{2}$ .
2. Graph the portion of the curve for $0\leq t\leq \dfrac{\pi }{2}$ .

$s=\dfrac{3b}{2}$
(b) b = 1

Arc Length
1. Find the arc length of the spiral $x(t) =t\cos t,\quad y(t) =t\sin t,$ $0\leq t\leq \pi$ .
2. Graph the portion of the curve for $0\leq t\leq \pi$ .

Distance Traveled In Problems 49–54, find the distance a particle travels along the given path over the indicated time interval.

$x(t)=3t,\quad y(t)=t^{2}-3;\; 0\leq t\leq 2$

$s = 5 + \dfrac{9}{4} \ln 3$

$x(t)=t^{2},\quad y(t)=3t;\;0\leq t\leq 2$

$x(t)=\dfrac{t^{2}}{2}+1,\quad \ y(t)=\dfrac{1}{3}(2t+3)^{3/2};\; 0\leq t\leq 2$

$s = \dfrac{3 \sqrt{11}-\sqrt{3}}{2} + \ln \left(\dfrac{3+\sqrt{11}}{1+\sqrt{3}}\right)$

$x(t)=a\cos t,\quad \ y(t)=a\sin t; \;a > 0, 0\leq t\leq \pi$

$x(t)=\cos (2t),\quad \ y(t)=\sin ^{2}t;\; 0\leq t\leq \dfrac{\pi }{2}$

$s=\sqrt{5}$

$x(t)=\dfrac{1}{t},\quad \ y(t)=\ln t;\;1\leq t\leq 2$

655

In Problems 55–57, find the speed at time $t$ of an object moving along each curve.

$x(t) =20t,\quad y(t) =-16t^{2}$

$\sqrt{400+1024t^{2}}$

$x(t) =t+\cos t,\quad y(t) =2t-\sin t$

$x(t) =20\sin ( 2t),\quad y(t) =6\cos t$

$\sqrt{1600\ \cos^{2}{(2t)}+36\ \sin^{2}{t}}$

Arc Length along a Circle Use the arc length formula for parametric equations to show that for a circle of radius $r,$ the length $s$ of the arc subtended by a central angle of $\theta$ radians is $s=r\theta .$

If the smooth curve $C$ , represented by the parametric equations $x=x(t), y=y(t),\quad \ a\leq t\leq b,$ is the graph of a function $y=f(x)$ , then $x$ can be used in place of the parameter $t,$ and the parametric equations for $C$ are $x=t,\quad y=f(t),\quad a\leq t\leq b.$ Show that the arc length formula for these parametric equations takes the form $\begin{equation*} s=\int_{a}^{b}\sqrt{\left( \frac{dx}{dt}\right) ^{2}+\left( \frac{dy}{dt} \right) ^{2}}\,dt=\int_{a}^{b}\sqrt{1+[ f' ( x) ] ^{2}}\,dx \end{equation*}$

See Student Solutions Manual.

Using Differentials to Approximate Arc Length Problems 60–63 use the following discussion:

For a smooth curve $C$ represented by the parametric equations $x=x(t),\quad y=y(t)$ , $a\leq t\leq b,$ the arc length $s$ satisfies the equation $\dfrac{ds}{dt}=\sqrt{\left( \dfrac{dx}{dt} \right) ^{2}+\left( \dfrac{dy}{dt}\right) ^{2}}$ , so that $\left( \dfrac{ds}{ dt}\right) ^{2}=\left( \dfrac{dx}{dt}\right) ^{2}+\left( \dfrac{dy}{dt} \right) ^{2}$ . In terms of differentials, this can be written as $\begin{eqnarray*} (ds)^{2}& =&(dx)^{2}+(dy)^{2} \\[4pt] ds& =&\sqrt{(dx)^{2}+(dy)^{2}} \end{eqnarray*}$

Geometrically, the differential $ds=\sqrt{(dx)^{2}+(dy)^{2}}$ is the length of the hypotenuse of a right triangle with sides of lengths $dx$ and $dy$ , as shown in the figure.

Because $\dfrac{dy}{dx}$ is the slope of the tangent line, the hypotenuse lies on the tangent line to the curve at $x.$ Then the differential $ds$ can be used to approximate the arc length $s$ between two nearby points. Using a differential to approximate arc length is particularly useful when it is difficult, or impossible, to find the arc length using $s=\int_{a}^{b}\sqrt{ \left( \dfrac{dx}{dt}\right) ^{2}+\left( \dfrac{dy}{dt}\right) ^{2}}\,dt.$

Use the differential ds to approximate each arc length.

$x(t) =t^{1/3},\;y(t) =t^{2}$ from $t=1$ to $t=1.1$

$x(t) =\sqrt{t},\;y(t) =t^{3}$ from $t=1$ to $t=1.2$

$0.73423$

$x(t) =a\sin t,\;y(t) =b\cos t$ from $t=0$ to $t=0.1$

$x(t) =e^{at},\;y(t) =e^{bt}$ from $t=0$ to $t=0.2$

$\sqrt{e^{0.4 a}-2e^{0.2 a}+e^{0.4 b}-2e^{0.2 b}+2}$

Challenge Problems

Show that a smooth curve $x=f(t),\;y=g(t)$ , for which $\dfrac{dx}{dt}$ is never $0$ , can be represented by a rectangular equation $y=F(x),$ where $F$ is differentiable. [Hint: Use the fact that $t=f^{-1}( x)$ exists and is differentiable.]

Find the point on the curve $x=\dfrac{4}{3}t^{3}+3t^{2},\;y=t^{3}-4t^{2}$ , for which the length of the curve from $(0,0)$ to $(x, y)$ is $\dfrac{80\sqrt{2}-40}{3}$ .

$(x(2),y(2)) =\left(\dfrac{68}{3},-8\right), (x(-2),y(-2)) =\left(\dfrac{4}{3},-24\right)$

Higher-Order Derivatives Find an expression for $\dfrac{ d^{2}y}{dx^{2}}$ if $x=f(t),\;y=g(t)$ , where $f$ and $g$ have second-order derivatives, and $\dfrac{dx}{dt}$ is never 0.

Find $\dfrac{d^{2}y}{dx^{2}}$ if $x( \theta) =a \cos ^{3}\theta ,\;y( \theta) =a \sin ^{3}\theta$ .

$\dfrac{1}{3a\ \cos^{4}{\theta}\ \sin{\theta}}$

Consider the cycloid defined by $x(t)=a(t-\sin t),\;y(t)=a(1-\cos t),\;a>0$ . Discuss the behavior of the tangent line to the cycloid at $t=0$ .

9.2 Assess Your UnderstandingPrinted Page 653

9.2 Assess Your Understanding
Printed Page 653